Question:
Evaluate the following limits
$\lim _{x \rightarrow 0} \frac{\tan \alpha x}{\tan \beta x}$
Solution:
To Find: Limits
NOTE: First Check the form of imit. Used this method if the limit is satisfied any one from 7 indeterminate form.
In this Case, indeterminate Form is $\frac{0}{0}$
Formula used: $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$.
So $\lim _{x \rightarrow 0} \frac{\tan \alpha x}{\tan \beta x}=\lim _{x \rightarrow 0}\left(\frac{\tan \alpha x}{\alpha x}\right) \times \frac{\beta x}{\sin \beta x} \times \frac{\alpha x}{\beta x}=\frac{\alpha x}{\beta x}=\frac{\alpha}{\beta}$
Therefore, $\lim _{x \rightarrow 0} \frac{\tan \alpha x}{\tan \beta x}=\frac{\alpha}{\beta}$